Rotation of a Solid Body round a Fixed Point. 333 



p, q, r being the components of the angular velocity w. The 

 position of the resultant couple may be expressed by means of 

 the ellipsoid (4). If a tangent plane be drawn to this ellipsoid 

 at the point (#, y, 2), and perpendicular to the line (a, |3, 7), it 

 may be easily shown that the projections of the triangle formed 

 by the radius vector and perpendicular are represented by the 

 quantities 



cos |3 cos 7 (I* - c 2 ), cos a cos 7 (c 2 - # 2 ), cos a cos J3 (a z - b*) : 



these three expressions multiplied by juw 2 will produce the quan- 

 tities used in (6). Hence it appears that the couple produced 

 by the centrifugal forces lies in the plane of the radius vector 

 and perpendicular to a tangent plane of the ellipsoid (4) ; the 

 tangent plane being perpendicular to the axis of rotation. Also, 

 the magnitude of the resultant couple is proportional to the 

 triangle formed by the radius vector and perpendicular. 



The differential equations of motion commonly used in the 

 solution of this problem may be deduced immediately from 

 equations (6). In fact, as the axes of co-ordinates are axes of 

 permanent rotation, the increment of angular velocity round 

 each axis will be equal to the statical couple of the applied forces 

 (including centrifugal forces), divided by the moment of inertia 

 round that axis. The statement of this fact, in analytical lan- 

 guage, will give the equations of motion : 



(7) 



(L, M, N) being the components of the applied statical couple. 

 The position and magnitude of the couple produced by the 

 centrifugal forces are easily found by the method which has been 

 just given ; but the direction will be found more readily by 



